Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-9x-3y &= -1 \\ -7x+y &= -3\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $3$ $\begin{align*}-9x-3y &= -1\\ -21x+3y &= -9\end{align*}$ Add the top and bottom equations. $-30x = -10$ Divide both sides by $-30$ and reduce as necessary. $x = \dfrac{1}{3}$ Substitute $\dfrac{1}{3}$ for $x$ in the top equation. $-9( \dfrac{1}{3})-3y = -1$ $-3-3y = -1$ $-3y = 2$ $y = -\dfrac{2}{3}$ The solution is $\enspace x = \dfrac{1}{3}, \enspace y = -\dfrac{2}{3}$.